We develop a Floer theoretical gluing technique and apply it
to deal with the most generic singular ber in the SYZ program,
namely the product of a torus with the immersed two-sphere with
a single nodal self-intersection. As an application, we construct
immersed Lagrangians in Gr(2;Cn) and OG(1;C5) and derive
their SYZ mirrors. It recovers the Lie theoretical mirrors constructed
by Rietsch. It also gives an e ective way to compute
stable disks (with non-trivial obstructions) bounded by immersed
Lagrangians.